A Mathematical Model of Liver Cell Aggregation In Vitro

The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

Weighted Block-Asynchronous Iteration on GPU-Accelerated Systems

Abstract. In this paper, we analyze the potential of using weights for block-asynchronous relaxation methods on GPUs. For this purpose, we introduce different weighting techniques similar to those applied in blocksmoothers for multigrid methods. For test matrices taken from the University of Florida Matrix Collection we report the convergence behavior and the total runtime for the different techniques. Analyzing the results, we observe that using weights may accelerate the convergence rate of block-asynchronous iteration considerably. While component-wise relaxation methods are seldom directly applied to systems of linear equations, using them as smoother in a multigrid framework they often provide an important contribution to finite element solvers. Since the parallelization potential of the classical smoothers like SOR and Gauss-Seidel is usually very limited, replacing them by weighted block-asynchronous smoothers may have a considerable impact on the overall multigrid performance. Due to the increase of heterogeneity in today’s architecture designs, the significance and the need for highly parallel asynchronous smoothers is expected to grow